## Linearly unrelated sequences.(English)Zbl 1005.11033

The author generalizes a result of P. Erdős [J. Math. Sci. 10, 1-7 (1975; Zbl 0372.10023)] on irrationality of sums of infinite series as follows: Let $$\{a_{i,n}\}_{n=1}^\infty$$ $$(i=1,\dots, K)$$ be sequences of positive real numbers. If for every sequence $$\{c_n\}_{n=1}^\infty$$ of positive integers the numbers $$\sum_{n=1}^\infty 1/(a_{1,n}c_n),\dots, \sum_{n=1}^\infty 1/(a_{K,n}c_n)$$, and 1 are linearly independent, then the sequences $$\{a_{i,n}\}_{n=1}^\infty$$ $$(i=1,\dots, K)$$ are linearly unrelated. Then the author proves:
Theorem. Let $$\{a_{i,n}\}_{n=1}^\infty$$, $$\{b_{i,n}\}_{n=1}^\infty$$ $$(i=1,\dots, K-1)$$ be sequences of positive integers and $$\varepsilon> 0$$ such that $\frac{a_{1,n+1}} {a_{1,n}}\geq 2^{K^{n-1}}, a_{1,n}|a_{1,n+1} \qquad (a_{1,n} \text{ divides }a_{1,n+1}), \tag{1}$
$b_{i,n}< 2^{K^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}, \qquad i=1,\dots, K-1, \tag{2}$
$\lim_{n\to\infty} \frac{a_{i,n} b_{j,n}} {b_{i,n} a_{j,n}}=0, \qquad \text{for all} \quad j,i\in \{1,\dots, K-1\},\;i>j, \tag{3}$
$a_{i,n} 2^{-K^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}< a_{1,n}< a_{i,n} 2^{K^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}, \qquad i=1,\dots, K-1, \tag{4}$ hold for every sufficiently large natural number $$n$$. Then the sequences $$\{a_{i,n}/ b_{i,n}\}_{n=1}^\infty$$ $$(i=1,\dots, K-1)$$ are linearly unrelated.
Further, let $$\{A_n\}_{n=1}^\infty$$ be a sequence of positive real numbers. If for every sequence $$\{c_n\}_{n=1}^\infty$$ of positive integers the series $$\sum_{n=1}^\infty \frac{1}{A_nc_n}$$ is irrational, then the sequence $$\{A_n\}_{n=1}^\infty$$ is irrational. If $$\{A_n\}_{n=1}^\infty$$ is not an irrational sequence, then it is a rational sequence. He proves
Theorem. Let $$\varepsilon>0$$, and let $$\{a_n\}_{n=1}^\infty$$ and $$\{b_n\}_{n=1}^\infty$$ be two sequences of positive integers such that $$a_n\geq 2^{2^n}$$ and $$b_n\leq 2^{2^{n-(\sqrt{2}+ \varepsilon) \sqrt{n}}}$$. Then the sequence $$\{\frac{\prod_{i=1}^n a_i}{b_n}\}_{n=1}^\infty$$ is irrational and the series $$\sum_{n=1}^\infty \frac{b_n} {\prod_{n=1}^n a_i}$$ is irrational too.
This theorem is an immediate consequence of the previous theorem. It is enough to put $$K=2$$. From the last theorem he also obtains a criterion for Cantor sequences to be irrational.

### MSC:

 11J72 Irrationality; linear independence over a field

### Keywords:

linearly unrelated sequences; irrational sequences

Zbl 0372.10023
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